Topologically flat embedded 2-spheres in specific simply connected 4-manifolds

TL;DR

This paper investigates the conditions under which certain homology classes in specific simply connected 4-manifolds can be represented by smooth or topologically flat embedded 2-spheres, addressing a fundamental question in 4-manifold topology.

Contribution

It provides new insights into the representability of homology classes by embedded spheres in particular simply connected 4-manifolds, advancing understanding of their topological and smooth structures.

Findings

Identifies conditions for representing homology classes by embedded spheres.

Distinguishes between smooth and topologically flat embeddings.

Contributes to the classification of 4-manifolds based on embedded sphere properties.

Abstract

In this note we study whether specific elements in the second homology of specific simply connected closed $4$-manifolds can be represented by smooth or topologically flat embedded spheres.